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OALib

Wave Functions, Creation and Annihilation Operators of Quantum Physical System

Malkhaz Mumladze

Published Jan 1, 2019• 2 citations
Publisher

Abstract

In this article we proposed derivation of the wave function of a quantum system from the functional representation of the state of this system.There is considered representation of creation and annihilation operators by boundary and co-boundary operators of chain and co-chain complexes on the physical space.
DOI
10.4236/oalib.1105166

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This is a self-contained and hopefully readable account on the method of creation and annihilation operators (also known as the Fock space representation or the "second quantization" formalism) for non-relativistic quantum mechanics of many particles. Assuming knowledge only on conventional quantum mechanics in the wave function formalism, we define the creation and annihilation operators, discuss their properties, and introduce corresponding representations of states and operators of many-particle systems. As the title of the note suggests, we cover most topics treated in sections 6.7 and 6.8 of Feynman's "Statistical Mechanics: A Set of Lectures". As a preliminary, we also carefully discuss the symmetry of wave functions describing indistinguishable particles. We note that all the contents of the present note are completely standard, and the definitions and the derivations presented here have been known to many. Although the style of the present note may be slightly more mathematical than standard physics literatures, we do not try to achieve full mathematical rigor.(Note to experts: In particular we here DERIVE the (anti)commutation relations of the creation and annihilation operators, rather than simply declaring them. In this sense our approach is quite close to that of Feynman's. But we here focus on the action of creation/annihilation operators on general NNN body wave functions, while Feynman makes a heavy use of Slater-determinant-type states from the beginning. We hope that our presentation provides a better perspective on the formalism.)
Operator algebra techniques are employed to derive the quantum evolution operator for the harmonic oscillator. The derivation begins with the construction of the annihilation and creation operators and the determination of the wave function for the coherent state as well as its time-dependent evolution, and ends with the transformation of the propagator in a mixed position-coherent-state representation to the desired one in configuration space. Throughout the entire procedure, besides elementary operator manipulations, it is only necessary to solve linear differential equations and to calculate Gaussian integrals.
Abstract The state vector of a quantized system of fields is defined as a functional of external source functions. Physical quantities are related to operators acting on such functionals. This corresponds to an over-all space-time view in which the state describes a space-time evolution and the state vector is not related to any special space-like surface. The field equations are derived by means of a simple formal quantization and are expressed as supplementary conditions restricting the state functionals. The equations are satisfied by generating functionals defined in the three-dimensional operator theory and were given previously. An investigation of the concept of functionals of anticommuting source functions leads immediately to the consideration of an antisymmetric tensor space with its creation and annihilation operators. The creation and annihilation operators introduced by Coester correspond to another representation and the connexion can be easily established. The relationship between different formal solutions of the field equations given by Coester, Nambu and Anderson, and Skyrme is made apparent. An interchange of the role of creation and annihilation operators, which amounts to a functional Fourier transformation, leads to an alternative description in which the state vector of the quantized system is given by a functional of fields which are functions of the four space-time co-ordinates.
The properties of a quantum mechanical system composed of many identical particles are most conveniently described in terms of the second-quantized, Heisenberg representation, particle-creation, and annihilation operators. The creation operator, ψ †( r , t), when acting to the right on a state of the system, adds a particle to the state at the space-time point r , t; the annihilation operator ψ ( r , t), the adjoint of the creation operator, acting to the right, removes a particle from the state at the point r , t.